I have that $$\int_{\mathbb{R}^N} \Phi(|u_n(x)|)dx\rightarrow \int_{\mathbb{R}^N}\Phi(|u(x)|) dx$$ where: $$u_n(x)\rightarrow u(x), \text{a.e.}$$ and $\Phi$ is a positive, continuous, convexe, increasing and an even function such that $$\exists K>0; \Phi(2u(x))\leq K\Phi(u(x)), \forall u .$$
I have to prove that $$\int_{\mathbb{R}^N} \Phi(|u_n(x)-u(x)|)dx\rightarrow 0 .$$
I tried this:
\begin{align}\int_{\mathbb{R}^N}\Phi(|u_n(x)-u(x)|)&=\int_{\mathbb{R}^N}\Phi(u_n(x)-u(x))\\&=\int_{\mathbb{R}^N}\Phi(\frac12(2(u_n(x)-u(x)))dx\\&\leq \int K \Phi(\frac12 u_n(x)+\frac12 u(x))dx\\&\leq \frac{K}{2}\left[\int\Phi(u_n(x))dx+\int_{\mathbb{R}^N}\Phi(u(x))dx\right] \end{align}
But this not give me the result because of the "+" I get in the final step.
How to do ?
Thank you